Swinging Waves in the Ablowitz-Ladik Equation

TL;DR

This paper introduces a new family of exact solutions for the Ablowitz-Ladik equation, featuring nonlinear phase dynamics and swinging wave behavior, expanding understanding of lattice wave phenomena.

Contribution

It presents novel cnoidal wave and soliton solutions with nonlinear phase dependence, derived via a two-point map, and explores their dynamics and quantized velocities.

Findings

New exact solutions with swinging phase behavior

Existence of standing and traveling waves with nonlinear phase

Explicit quantization rule for wave velocity

Abstract

We construct a novel family of exact cnoidal wave and soliton solutions of the focusing and defocusing Ablowitz-Ladik equations. Unlike cnoidal waves that were obtained by earlier authors, the phase variable of the new solutions exhibits a nonlinear dependence on time and site number; the wave ``swings". Our approach hinges on the existence of a two-point map governing the absolute value of the complex field; this map gives rise to standing waves centred arbitrarily relative to the lattice sites. Having derived stationary solutions, we use these as a basis for constructing waves with nonzero velocity. The localised members of the new family comprise dark solitons with the nontrivial asymptotic behaviour. We identify periodic and quasiperiodic patterns and establish an explicit quantisation rule for the velocity of the wave circulating in a closed loop of $N$ sites.